Multiplicative sequence

In mathematics, a multiplicative sequence or m-sequence is a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology.

In mathematics, a multiplicative sequence or m-sequence is a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology. ==Definition== Let K n be polynomials over a ring A in indeterminates p 1 , ... weighted so that p i has weight i (with p 0 = 1) and all the terms in K n have weight n (in particular K n is a polynomial in p 1 , ..., p n ). The sequence K n is multiplicative if the map K: \sum_{n = 0}^\infty q_nz^n\mapsto \sum_{n=0}^\infty K_n(q_1,\cdots, q_n)z^n is an endomorphism of the multiplicative monoid (A[x_1, x_2,\cdots]z,\cdot) , where q_n\in A[x_1, x_2,\cdots] . The power series K(1+z) = \sum K_n(1,0,\ldots,0) z^n is the characteristic power series of the K n . A multiplicative sequence is determined by its characteristic power series Q(z), and every power series with constant term 1 gives rise to a multiplicative sequence. To recover a multiplicative sequence from a characteristic power series Q(z) we consider the coefficient of z  j in the product \prod_{i=1}^m Q(\beta_i z) \ for any m > j. This is symmetric in the β i and homogeneous of weight j: so can be expressed as a polynomial K j (p 1 , ..., p j ) in the elementary symmetric functions p of the β. Then K j defines a multiplicative sequence. ==Examples== As an example, the sequence K n = p n is multiplicative and has characteristic power series 1 + z. Consider the power series Q(z) = \frac{\sqrt z}{\tanh \sqrt z} = 1 - \sum_{k=1}^\infty (-1)^k \frac{2^{2k}}{(2k)!} B_k z^k \ where B k is the k-th Bernoulli number. The multiplicative sequence with Q as characteristic power series is denoted L j (p 1 , ..., p j ). The multiplicative sequence with characteristic power series Q(z) = \frac{2\sqrt z}{\sinh 2\sqrt z} \ is denoted A j (p 1 ,...,p j ). The multiplicative sequence with characteristic power series Q(z) = \frac{z}{1-\exp(-z)} = 1 + \frac{x}{2} - \sum_{k=1}^\infty (-1)^k \frac{B_k}{(2k)!} z^{2k} \ is denoted T j (p 1 ,...,p j ): these are the Todd polynomials. ==Genus== The genus of a multiplicative sequence is a ring homomorphism, from the cobordism ring of smooth oriented compact manifolds to another ring, usually the ring of rational numbers. For example, the Todd genus is associated to the Todd polynomials with characteristic power series \frac{z}{1-\exp(-z)} . ==References==

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